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Paul Malliavin Anton Thalmaier
Stochastic Calculus
of Variations
in Mathematical
Finance
ABC
PaulMalliavin AntonThalmaier
AcadémiedesSciences DépartementdeMathématiques
InstitutdeFrance UniversitédePoitiers
E-mail:[email protected] E-mail:[email protected]
MathematicsSubjectClassification(2000):60H30,60H07,60G44,62P20,91B24
LibraryofCongressControlNumber:2005930379
ISBN-10 3-540-43431-3SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-43431-3SpringerBerlinHeidelbergNewYork
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orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
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Dedicated to Kiyosi Itoˆ
Preface
Stochastic Calculus of Variations (or Malliavin Calculus) consists, in brief,
in constructing and exploiting natural differentiable structures on abstract
probabilityspaces;inotherwords,StochasticCalculusofVariationsproceeds
from a merging of differential calculus and probability theory.
Asoptimizationunderarandomenvironmentisattheheartofmathemat-
ical finance, and as differential calculus is of paramount importance for the
search of extrema, it is not surprising that Stochastic Calculus of Variations
appears in mathematical finance. The computation of price sensitivities (or
Greeks) obviously belongs to the realm of differential calculus.
Nevertheless, Stochastic Calculus of Variations was introduced relatively
late in the mathematical finance literature: first in 1991 with the Ocone-
Karatzas hedging formula, and soon after that, many other applications ap-
peared in various other branches of mathematical finance; in 1999 a new im-
petus came from the works of P.L.Lions and his associates.
Ourobjectivehasbeentowriteabookwithcompletemathematicalproofs
together with a relatively light conceptual load of abstract mathematics; this
point of view has the drawback that often theorems are not stated under
minimal hypotheses.
To faciliate applications, we emphasize, whenever possible, an approach
through finite-dimensional approximation which is crucial for any kind of nu-
merical analysis. More could have been done in numerical developments (cal-
ibrations, quantizations, etc.) and perhaps less on the geometrical approach
to finance (local market stability, compartmentation by maturities of interest
rate models); this bias reflects our personal background.
Chapter1and,tosomeextent,partsofChap.2,aretheonlyprerequisites
toreadingthisbook;theremainingchaptersshouldbereadableindependently
of each other. Independence of the chapters was intended to facilitate the
access to the book; sometimes however it results in closely related material
being dispersed over different chapters. We hope that this inconvenience can
be compensated by the extensive Index.
The authors wish to thank A. Sulem and the joint Mathematical Finance
group of INRIA Rocquencourt, the Universit´e de Marne la Vall´ee and Ecole
Nationale des Ponts et Chauss´ees for the organization of an International
VIII Preface
Symposium on the theme of our book in December 2001 (published in Math-
ematical Finance, January 2003). This Symposium was the starting point for
our joint project.
Finally, we are greatly indepted to W. Schachermayer and J. Teichmann
for reading a first draft of this book and for their far-reaching suggestions.
Lastnotleast,weimplorethereadertosendanycommentsonthecontentof
thisbook,includingerrors,[email protected],
so that we may include them, with proper credit, in a Web page which will
be created for this purpose.
Paris, Paul Malliavin
April, 2005 Anton Thalmaier
Contents
1 Gaussian Stochastic Calculus of Variations................. 1
1.1 Finite-Dimensional Gaussian Spaces,
Hermite Expansion ...................................... 1
1.2 Wiener Space as Limit of its Dyadic Filtration .............. 5
1.3 Stroock–Sobolev Spaces
of Functionals on Wiener Space ........................... 7
1.4 Divergence of Vector Fields, Integration by Parts ............ 10
1.5 Itoˆ’s Theory of Stochastic Integrals ........................ 15
1.6 Differential and Integral Calculus
in Chaos Expansion...................................... 17
1.7 Monte-Carlo Computation of Divergence ................... 21
2 Computation of Greeks
and Integration by Parts Formulae ........................ 25
2.1 PDE Option Pricing; PDEs Governing
the Evolution of Greeks .................................. 25
2.2 Stochastic Flow of Diffeomorphisms;
Ocone-Karatzas Hedging ................................. 30
2.3 Principle of Equivalence of Instantaneous Derivatives ........ 33
2.4 Pathwise Smearing for European Options................... 33
2.5 Examples of Computing Pathwise Weights.................. 35
2.6 Pathwise Smearing for Barrier Option...................... 37
3 Market Equilibrium and Price-Volatility Feedback Rate ... 41
3.1 Natural Metric Associated to Pathwise Smearing ............ 41
3.2 Price-Volatility Feedback Rate ............................ 42
3.3 Measurement of the Price-Volatility Feedback Rate .......... 45
3.4 Market Ergodicity
and Price-Volatility Feedback Rate ........................ 46
X Contents
4 Multivariate Conditioning
and Regularity of Law ..................................... 49
4.1 Non-Degenerate Maps.................................... 49
4.2 Divergences............................................. 51
4.3 Regularity of the Law of a Non-Degenerate Map............. 53
4.4 Multivariate Conditioning ................................ 55
4.5 Riesz Transform and Multivariate Conditioning ............. 59
4.6 Example of the Univariate Conditioning.................... 61
5 Non-Elliptic Markets and Instability
in HJM Models............................................ 65
5.1 Notation for Diffusions on RN............................. 66
5.2 The Malliavin Covariance Matrix
of a Hypoelliptic Diffusion ................................ 67
5.3 Malliavin Covariance Matrix
and Ho¨rmander Bracket Conditions........................ 70
5.4 Regularity by Predictable Smearing........................ 70
5.5 Forward Regularity
by an Infinite-Dimensional Heat Equation .................. 72
5.6 Instability of Hedging Digital Options
in HJM Models ......................................... 73
5.7 Econometric Observation of an Interest Rate Market......... 75
6 Insider Trading ............................................ 77
6.1 A Toy Model: the Brownian Bridge ........................ 77
6.2 Information Drift and Stochastic Calculus
of Variations............................................ 79
6.3 Integral Representation
of Measure-Valued Martingales............................ 81
6.4 Insider Additional Utility................................. 83
6.5 An Example of an Insider Getting Free Lunches ............. 84
7 Asymptotic Expansion and Weak Convergence ............ 87
7.1 Asymptotic Expansion of SDEs Depending
on a Parameter ......................................... 88
7.2 Watanabe Distributions and Descent Principle .............. 89
7.3 Strong Functional Convergence of the Euler Scheme ......... 90
7.4 Weak Convergence of the Euler Scheme .................... 93
8 Stochastic Calculus of Variations for Markets with Jumps . 97
8.1 Probability Spaces of Finite Type Jump Processes ........... 98
8.2 Stochastic Calculus of Variations
for Exponential Variables.................................100
8.3 Stochastic Calculus of Variations
for Poisson Processes ....................................102
Contents XI
8.4 Mean-Variance Minimal Hedging
and Clark–Ocone Formula ................................104
A Volatility Estimation by Fourier Expansion ................107
A.1 Fourier Transform of the Volatility Functor .................109
A.2 Numerical Implementation of the Method ..................112
B Strong Monte-Carlo Approximation
of an Elliptic Market ......................................115
B.1 Definition of the Scheme S ...............................116
B.2 The Milstein Scheme.....................................117
B.3 Horizontal Parametrization ...............................118
B.4 Reconstruction of the Scheme S ..........................120
C Numerical Implementation
of the Price-Volatility Feedback Rate ......................123
References.....................................................127
Index..........................................................139
